eslHQ - no/any

"No" = " Not ... any", so any time you could use one you could, grammatically, use the other. Eg -

There isn't any milk in the fridge = There's no milk in the fridge.
We did n't see any sign of bats in the cave = We saw no sign of bats in the cave

However, the use of "no" is often a marker of more formal speech, so that except in certain cases it's more common in conversation to use "not ...any". The exceptions are with the verb Be - as in example 1 - and in certain semi-idiomatic expressions like "no sign of " expressions like "no sign of" in example 2 above. Other common examples are I've no idea, There's no hope, I've no doubt, There's no problem, it's no wonder ..etc. Oddly, when "no" is used with the verb Have it tends to break the normal grammatical rules and occur in contracted form - despite the fact that it's a main verb. This is why I'd consider it idiomatic.

"No", like "not" and "never", is a negative word in its own right. It simply negates the noun rather than the verb. But I'd argue that the resulting sentence is negative.

The fact that it negates the noun also means that a noun phrase with no can also be used as the subject of a sentence : No self-respecting person could believe that.... Or just on it's own : no woman no cry.

As far as teaching it, I'd concentrate on not ...any first, and then when it comes up point out the relationship between the two - with the caveat that no is trickier to use being either stylistically marked or idiomatic. The type of exercises found in Murphy's Essential English Grammar are good practice activities.